Finite Group Databasebeta

← #134830

Subgroups — #134830

Order 512.

336 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22212C211yesmaximal, characteristic
2#11531C212yesmaximal
3#8612C211yesmaximal, characteristic
4#8611C211yesmaximal, characteristic
5#7249C212yesmaximal
6C2×#446C411yescharacteristic
7C2×C8×D4C2211yescharacteristic
8C4×#124C2211yescharacteristic
9#1984C411yescharacteristic
10#1954C2211yescharacteristic
11#1924—21no—
12#1924—21no—
13#1850—21no—
14#1850—21no—
15#1842C411yescharacteristic
16#1810—21no—
17#1810—21no—
18#1744C411yescharacteristic
19#1465C2212yes—
20#1465C2212yes—
21#1324—22no—
22#1319—22no—
23#1293—21no—
24#1293—21no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#124Q811yescharacteristic
28C2×C4×Q8C2×C411yescharacteristic
29C2×C4×D4C2311yescharacteristic
30#446—41no—
31#446—41no—
32C8×D4—41no—
33C8×D4—41no—
34C2×#108—41no—
35C2×#108—41no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C4⋊C8C2×C411yescharacteristic
41#407—22no—
42#407—22no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8D411yescharacteristic
46C2×C22⋊C8D411yescharacteristic
47C2×#97—21no—
48C2×C4×C8C2×C411yescharacteristic
49C2×C4×C8C2×C411yescharacteristic
50#391D411yescharacteristic
51#390Q811yescharacteristic
52#388—21no—
53#388—21no—
54#382D411yescharacteristic
55C4×C22⋊C4—21no—
56C4×C22⋊C4—21no—
57C43C2×C411yescharacteristic
58#341—41no—
59#341—41no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#324—22no—
65#324—22no—
66#323—22no—
67#323—22no—
68C22×Q8C4⋊C411yescharacteristic
69C22×D4C2×Q811yescharacteristic
70C23×C4C2×D411yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×M16—42no—
73C2×M16—42no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—81no—
81C22×C8—81no—
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22⋊C411yescharacteristic
84C22×C8C22⋊C411yescharacteristic
85C22×C8C22⋊C411yescharacteristic
86#124—42no—
87C4×Q8—41no—
88C4×Q8—41no—
89C4×D4—41no—
90C4×D4—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C4⋊C411yescharacteristic
93C2×C4⋊C4C2×D411yescharacteristic
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4#4111yescharacteristic
99C2×C22⋊C4#4111yescharacteristic
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42C4⋊C411yescharacteristic
104C2×C42C4⋊C411yescharacteristic
105C2×C42C22×C411yescharacteristic
106#108—82no—
107#108—82no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111C4⋊C8—41no—
112C4⋊C8—81no—
113C4⋊C8—81no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—42no—
117C22⋊C8—42no—
118C22⋊C8—42no—
119C22⋊C8—42no—
120C22⋊C8—81no—
121C22⋊C8—81no—
122#97—42no—
123C4×C8—21no—
124C4×C8—21no—
125C4×C8—41no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130C24C22⋊Q811yescharacteristic
131C24C22⋊Q811yescharacteristic
132C2×Q8—22no—
133C2×Q8—41no—
134C2×Q8—41no—
135C2×D4—22no—
136C2×D4—41no—
137C2×D4—41no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4C2×C22⋊C411yescharacteristic
151C22×C4C2×C4⋊C411yescharacteristic
152C22×C4#11711yescharacteristic
153C22×C4#12111yescharacteristic
154C22×C4#12111yescharacteristic
155M16—82no—
156M16—82no—
157M16—82no—
158M16—82no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8—42no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—82no—
177C2×C8—82no—
178C2×C8#9911yescharacteristic
179C2×C8#9911yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42—21no—
190C42—21no—
191C42—22no—
192C42—41no—
193C42—41no—
194C42—41no—
195C42—42no—
196C42—41no—
197C42—41no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23#38011yescharacteristic
209Q8—82no—
210D4—44no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4C2×#9911yescharacteristic
240C2×C4#41011yescharacteristic
241C2×C4#43012yes—
242C8—21no—
243C8—21no—
244C8—41no—
245C8—41no—
246C8—41no—
247C8—81no—
248C8—81no—
249C8—162no—
250C8—162no—
251C22—21no—
252C22—21no—
253C22—42no—
254C22—42no—
255C22—41no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—81no—
263C22#179911yescharacteristic
264C22#181111yescharacteristic
265C22#181311yescharacteristic
266C4—21no—
267C4—22no—
268C4—42no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—81no—
273C4—81no—
274C4—81no—
275C4#212711yescharacteristic
276C4#212911yescharacteristic
277C2—21no—
278C2—21no—
279C2—81no—
280C2—81no—
281C2#861311yescharacteristic
282C2#1280011yescharacteristic
283C2#1280911yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC2×C42C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC22×D4C2×Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#222122
#115312
#86122
#86112
#72492